1. Field of the Invention
The present invention generally relates to Very Large Scale Integrated (VLSI) circuit fabrication, and more particularly, to a method and system for determining numerical and discretization errors into a pixel based image simulation used in lithographic mask design.
2. Description of the Related Art
Considerable efforts have been made in the field of lithographic methods used in the manufacture of integrated circuits. The con design of masks used in optical lithographic y to develop and implement methods of mask compensation and verification, e.g., model-based optical proximity correction (MBOPC). MBOPC is integral to the mask design process, in which initial mask shapes are modified or “corrected” to compensate for distortions in the imaging process, as known in the art. The lithographic process model will include a model of the optical effects, and may also include a model of the resist process and other processes, such as the etch process. In MBOPC, the lithographic model is used to simulate the image of the mask shapes, and then the simulated images are compared to the target shapes desired to be printed on the wafer. If the differences between the simulated images and target shapes exceed predetermined criteria, then the mask shapes are modified and the process is repeated using the modified mask layout until the simulated images match the target shapes within acceptable tolerances.
Typically, the model images have been computed at evaluation points representing edge segments of mask shape edges. Fragment-based image simulations involve computing the image at or in the vicinity of a set of selected evaluation points to represent an edge fragment of mask feature polygons which, for example, are adjusted during OPC. Such edge fragments are typically unevenly spaced. However, as IC technology drives towards smaller and denser layouts, the computational cost of simulating images is becoming increasingly impractical.
An alternative method for performing lithographic image simulation is pixel-based simulation. A key aspect of this simulation approach is that the images of the projected mask patterns are calculated on a regular grid over the entire layout, which permits the use of more computationally efficient methods, such as Fourier transforms. However, the image contours that are defined by interpolating image intensity curves between pixel grid points.
In the OPC tool, the modification of mask shape edges are performed by adjusting the location of edge fragments in increments that are defined by the requirements of the mask manufacturing and inspection tools. These increments are referred to hereinafter as the OPC grid. The OPC grid element width A is the smallest line segment, according to mask manufacturability requirements, that is used to fragment and modify shape edges during the OPC process. The resolution of the OPC grid depends on the capabilities of the mask inspection tool and mask manufacturing tool. The mask shapes will be aligned to the OPC grid. For 45 nm technology, the OPC grid size is typically 0.25 nm, resulting in about 180 OPC grid elements per CD.
The advantage of pixel-based imaging relative to fragment-based imaging is explained in FIGS. 1A through 1D. FIG. 1A illustrates a rectangular mask shape 110 in a layout 105. In this example of fragment-based simulation, the optical and the resist image is simulated at 6 evaluation locations 111 as well as along cut-lines 103 running through each evaluation point 111. The number of image simulation points along each cut-line typically needed is about 15 image simulations for the complete optical and the resist image simulation. Therefore, for the case in 101, the number of points at which image would be simulated is 90 for six cut-lines.
The same layout 105 with the shape 110 in it is shown again in FIG. 1B in a pixel-based imaging method. In the pixel-based imaging method, the layout 105 is divided into uniform pixels 112 and image intensity is typically simulated at a point, e.g. the center point that represents the intensity of the pixel. In this example, image is simulated at 225 points. The pixels typically each have a uniform size along two orthogonal directions, x and y, but which need not be the same size in each direction. The selection of the size of the pixel grid elements is based on the effective resolution of the lithographic process, which is related to the Raleigh limit given by λ/(4NA), where λ is the wavelength of the illumination energy, NA is the numerical aperture. The pixel grid is given by α=λ/(k4NA) where k is a scaling or oversampling factor. For 45 nm technology, λ=193 nm. The numerical aperture of the optical system is typically about 1.2, but may range from about 0.45 to 1.3. The scaling or oversampling factor k is related to the development process, for example, the resist or etch processes. For a purely optical process, k=1. For chemically amplified resists, k is typically between about 1.5 to 2. The value of k is positive may be selected based on a speed-accuracy tradeoff. Large values of k can increase the accuracy of interpolation of the image values between computed pixel grid points but significantly add to the cost of computation. A pixel grid with k=1 represents the smallest unit at which the lithographic process can respond to a change in design information. Thus, for the case of 45 nm technology, using λ=193 nm, k=2, and NA=1.2, the size of a pixel grid element a may be chosen to be about 20 nm.
As feature size gets smaller with coming technologies, more and more shapes are accommodated in the same area of the mask. This is illustrated in FIGS. 1C and 1D. The layout 107 with the same area as layout 105 now accommodates three shapes 120. In this example of fragment-based imaging computation, as shown in FIG. 1C, there are 10 cut lines per shape totaling 30 cut lines. Assuming again there are 15 computation points needed per cut line, the fragment-based simulation would require a total of 450 image computations for the fragment-based simulation. On the other hand for the pixel-based image simulation, the same gridding may be used as shown in FIG. 1D, so that the number of image computations still remains 225, which is considerably smaller than the number of computations needed for the fragment-based simulation in FIG. 1C.
FIG. 1E to 1F further elaborates the above concept. In FIG. 1E, a mask shape 150, in this example, a rectangle having width 151 and length 152. Width 151 may be as small as the minimum resolvable width w1 and the aspect ratio of length 152 to width is typically at least 4-5. For executing the fragment based Optical Proximity Correction, the edges of shape 150 is broken up into several edge segments 156. The end points 153 define the segments 156. The image corresponding to a segment 156 is typically evaluated at an evaluation point 154 that represents the image of the entire segment 156. Typically, the evaluation point 154 is located at the midpoint of a segment 156, but may be staggered, e.g. near corners of shapes. In resist process models, optical image characteristics besides image intensity at the evaluation point are also required. In order to compute these additional image characteristics, the optical image is typically computed along a cut-line 155 that is drawn through the evaluation point 154 which is typically orthogonal to the line segment 156. In the case of segments located near a corner, the cut-line 155 may be oriented along a non-orthogonal direction relative to the edge segment. The number of image computation points along the cut line needed to compute the resist image is about 10 to 20 points (typically 15) along the cut-line 155, which may be either uniformly or non-uniformly spaced. The length of a cut-line 155 is typically in the range 0.75 w1 to 1.5 w1.
After several iterations of OPC, the initial mask shape 150 has been modified as shape 160 (see FIG. 1F). The original segments 156 have been moved by the OPC tool to segments 166 in modified shape 160. The modified segments 166 are defined by end points 163 whose locations have been modified from the original locations 153. In the same manner, the image evaluation points 154 are modified to result in image evaluation points 164 after the OPC iterations, and the cut-lines 155 are modified to result in new cut-lines 165. The movement of the end points 153 to modified end points 163 is performed in increments that are along the smallest grid that is defined by the mask manufacturing capabilities. This minimum step size for modifying mask shapes in an OPC tool is referred to hereinafter as the “OPC grid”. For example, the movement of an edge segment 156 to segment 166 is along a distance 161 that is a multiple of this smallest grid, which is defined as the OPC grid. For example, for 45 nm technology, the OPC grid size is 0.25 nm.
The computation of the image in a pixel based image computation is illustrated in FIG. 2A. The modified shape 160 resulting from OPC will be used to simulate the image, shown overlain on a pixel grid 200, comprising pixel elements 170. The original shape 150 is illustrated for reference. In case of pixel based simulation, the image is evaluated at the pixel grid points 171, rather than along a fragment cut-line as in the fragment-based simulation. The simulated printed image is typically represented by a contour line 180 that plots the constant image intensity that corresponds to the print threshold of the lithographic process. The values for contour 180 are computed by appropriately interpolating between the image values obtained at the pixel grid points 171.
The change of pixel size as a function of technology is illustrated in FIG. 2B. It shows that pixel size decreases as node size shrinks, but at a slower rate as node size shrinks. The size of the pixel grid is typically changed with different technology node to ensure the grid will adequately provide information about the image. For 65 nm through 22 nm, the wave length of light λ is expected to remain at 193 nm. The numerical aperture (NA) is expected to grow slowly from 0.8 to 1.45. Therefore, the pixel-grid almost remains the same compared to the change in the feature size (FIG. 2B).
Clearly from the above examples, as technology advances towards smaller features, pixel-based image computation becomes more and more efficient than fragment-based image computations. Referring to FIG. 2C, the number of points required for image computation is shown for fragment-based imaging, for cases using both tighter and more relaxed pitch, and compared to the number of points required in pixel-based imaging. As the technology continues to shrink, the number of computation points required in fragment-based image simulation increases more rapidly relative to pixel-based image simulation. For 45 nm technology and smaller pixel-based imaging computation becomes more efficient than fragment-based image computation.
A flow chart describing an OPC iteration in pixel-based imaging is shown in FIG. 3. The regularity of pixel-based imaging grid lends itself to fast Fourier transform (FFT) based computations (e.g. convolutions). FFT-based convolutions are relatively rapid compared to the classic convolution algorithms used in the earlier unevenly sampled fragment-based simulation, assuming that current practice requires calculating the partially coherent images on a uniformly sampled basis. Fast convolutions typically involve both FFT-s and inverse FFT-s, but since FFT-s and inverse FFT-s are calculated in almost exactly the same way, we will generally use the term “FFT” to refer to both.
Block 301 in FIG. 3 shows a mask layout containing geometric shapes. In Block 302 the mask layout is rasterized or divided into pixels to create a rastered or pixelized mask image as shown in Block 303. In Block 304 the rasterized mask layout is FFT-ed to create a mask image in the spatial frequency domain as shown in Block 305. In Block 306 the mask layout in frequency domain is convolved with optical and resist kernels (which amounts to point-wise multiplication) to create a convolved mask image. In Block 307 the convolved mask layout in the frequency domain is inverse-FFT-ed to create image contour (Block 308) back in the spatial domain. Finally in block 309 the mask edges in the spatial domain are moved based on the current image contour and the cycle is repeated for the next iteration of OPC. The point wise multiplication and FFT adds to the computational efficiency of the pixel-based imaging method.
Besides the computational advantages of pixel-based imaging, as described above, the pixel-based imaging is still preferable for various reasons. For example, calculation of image on a dense uniform grid enables more robust resist models in simulation, since the full exposure distribution in the neighborhood of each developed edge is available to the model as a complete and physically realistic input. Also, the regularity of pixel-based imaging is better suited to parallel computation than is non-uniform fragment-based image sampling, and this regularity provides a predictability during algorithm flow that special-purpose computer hardware can optionally exploit to improve the efficiency of CPU utilization, for example by reducing memory latency.
Although dense pixel-based image computation is computationally efficient, some of the steps described in FIG. 3 are inherently prone to a variety of errors due to numerical and discretization. This is illustrated in FIG. 4. For example, rasterization (described as 302 in FIG. 3) can result in discretization errors. The sampling to Nyquist grid (401) can have problems such as under-sampling. Both FFT (304) and Inverse FFT (307) can have problems such as numerical errors or frequency errors. Convolution (306) can also have numerical errors.
There are methods known to attempt to correct for these errors. Some of these examples are shown in FIG. 4. The discretization can be reduced by anti-aliasing. The under-sampling can be reduced by proper interpolation or careful over-sampling. The numerical errors can be reduced by high precision implementation. The frequency errors can be prevented by providing a guard-band. Some methods of compensating for such numerical or discretization errors may introduce computational inefficiencies, and it would be preferable to avoid them if possible.
Thus, there may be many variations in the implementation of pixel-based image simulations. However, in designing and calibration of the grid for a pixel-based simulation, it may be difficult to determine whether inaccuracies in a specific grid implementation are due to discretization or numerical errors or errors in the approximations used in the physical representation of the lithographic process.
In view of the foregoing, there is a need for a method to determine and minimize numerical and discretization errors in pixel-based imaging separately from other, more physically-based errors.